Here are some common issues with Godox firmware updating:
Mismatching Firmware File
Whatever you do, DO NOT USE A FIRMWARE FILE FOR A DIFFERENT DEVICE. This is a basic tenet of all firmware updates: firmware is closely tied to hardware configurations, and any hardware differences can mean bricking a device if you overwrite with the wrong firmware image.
While a TT685 and V860II might be identical in UI and features, the fact that they both use different power sources means the firmware images are not identical. Each device has its own separate firmware file for a reason. TT350 users found this one out the hard way when Adorama mistakenly posted the TT350-C update (specifically for adding compatibility with some newer EOS M bodies) as a generic TT350 update. So many non-Canon TT350 owners bricked their flashes with this update, Godox eventually posted the initial versions of the TT350 firmware.
If your device does not yet have a firmware update posted, suck it up and wait until it does.
The only way to fix a mismatch like this is to reload the proper firmware image, but this depends on the device being able to recognize the firmware update request, and sometimes the correct/current firmware image you need is not available. Requesting the firmware from your retailer or Godox may be your only recourse.
G1 or G2
Godox has two separate firmware updating apps. G1 is for pre-AD200 devices, G2 is for AD200 and later devices. Check the Godox downloads page to make sure you're using the correct application for your device. Both applications, when installed, should also install a custom USB driver.
Cable and port
Make sure you're using the correct cable for the device. Newer devices (e.g., A1, Xpro, R2 Pro II, AD400 Pro, X2T) use a USB-C cable, while older devices (e.g., TT685, V860II, AD200) use a micro USB cable. If you are updating an AD200, AD360II, or AD600, make sure that you are using the micro USB port, not the USB-A port that is for attaching an FTR16 receiver if you want to use Godox's older 433 MHz radio system.
If you are using a micro USB cable and the G1/G2 software doesn't recognize the device, consider trying a different cable. Some micro USB cables are solely used for charging and do not do data transfer.
If you downloaded your files from the Godox site, rather than the Flashpoint one, pathnames of either the updates or the applications may contain two-byte Chinese characters which can cause an error. Removing the two-byte characters fixes the issue.
The downloaded file is a .rar archive that contains the firmware update file, and a PDF that contains the revision history for the firmware. If Internet Explorer renamed it as a .man archive and you do not have tools to extract the archive contents, you can simply rename it to .rar.
Also, the .rar format has changed in recent updates so you may simply need to update your archiving application (e.g., update Winzip to the latest version). Or you can use the latest version of the open source 7Zip.
Disable driver signing
Someone on dpreview mentions that you may also need to disable driver signing to get the updater to work.
See also: Flash Havoc's article on the G1 app.